ECAL'95: 3rd. European Conference on Arti cial Life, F. Moran, A. Moreno, J. J. Merelo, and P. Chacon (eds.), pages 544-554, 1995. Copyright Springer-Verlag 1995.
Quasi-Uniform Computation-Universal Cellular Automata Moshe Sipper Department of Computer Science Tel Aviv University Tel Aviv 69978, Israel e-mail:
[email protected]
Abstract. Cellular automata (CA) are dynamical systems in which space and time are dis-
crete, where each cell obeys the same rule and has a nite number of states. In this paper we study non-uniform CA, i.e. with non-uniform local interaction rules. Our focal point is the issue of universal computation, which has been proven for uniform automata using complicated designs embedded in cellular space. The computation-universal system presented here is simpler than previous ones, and is embedded in the minimal possible two-dimensional cellular space, namely 2-state, 5-neighbor (which is insucient for universal computation in the uniform model). The space studied is quasi-uniform, meaning that a small number of rules is used (our nal design consists of just two rules which is minimal), distributed such that most of the grid contains one rule except for an in nitely small region which contains the others. We maintain that such automata provide us with a simple, general model for studying Arti cial Life phenomena.
1 Introduction Cellular automata (CA) are dynamical systems in which space and time are discrete. The states of cells in a regular grid are updated synchronously according to a local interaction rule. Each cell obeys the same rule and has a nite (usually small) number of states [29]. The model was originally conceived by John von Neumann in the 1950's to provide a more realistic framework for studying the behavior of complex, extended systems [31]. Over the years it has been applied to the study of general phenomenological aspects of the world, including: communication, computation, construction, growth, reproduction, competition and evolution [4, 26, 29]. The CA model is both general and simple. Generality implies two things: (1) the model supports universal computation and (2) the basic units encode a general form of interaction rather than some specialized action. Simplicity implies that the basic units of interaction are modest in comparison to Turing machines. If we imagine a scale of complexity with Turing machines occupying the high end then simple machines are those that occupy the low end, e.g. nite state automatons. The CA model is one of the simplest, general models available. From an Arti cial Life (AL) perspective it has been noted that the main diculty with the CA approach seems to lie with the extreme low-level representation of the interactions. CA's are programmed at the level of the local physics of the system and, therefore, higher-level cooperative structures are dicult to evolve in CA's [23]. One of our primary goals is to increase the \capacity" for AL modeling while preserving the essential features of the original CA, namely massive parallelism, locality of cellular interactions and simplicity of cells. Thus we attain a model that is close to CA with regards to generality and simplicity [24, 25]. In this paper we consider non-uniform CA, i.e. with non-uniform local interaction rules. Such automata function in the same way as uniform ones, the only di erence being in the cellular rules which need not be identical for all cells. This type of model has been investigated by others. [8] presents two generalizations of cellular automata, namely discrete neural networks and automata networks. These are compared to the
original model from a computational point of view which considers the classes of problems such models can solve. The work of [30] discusses a one-dimensional CA in which a cell probabilistically selects one of two rules, at each time step. They showed that complex patterns appear characteristic of class IV behavior1 , however universal computation was not discussed (see also [9]). In [13, 22] adaptive stochastic cellular automata are considered which are essentially non-uniform automata whose rules are drawn from the same probability distribution function. Their approach focuses on the learning aspect where an automaton is trained to solve some problem (e.g. pole balancing). As noted above, our motivation is di erent than these works: we wish to study Arti cial Life phenomena such as evolution, emergence and multi-cellularity in a simple, general model which operates at a higher level than CA [24, 25]. The focal point of this paper is the issue of universal computation in two-dimensional CA, namely the construction of machines, embedded in cellular space, whose computational power is equivalent to that of a universal Turing machine [10]. The rst such machine was described by von Neumann, who used 29 state, 5-neighbor cells [31]2. Codd provided a detailed description of a computer embedded in an 8-state, 5-neighbor cellular space, thus reducing the complexity of von Neumann's machine [5]. Banks described a 2-state and a 3-state automaton (both 5-neighbor) which support universal computation with an in nite and nite initial con guration3, respectively [1]. A cellular space with a minimal number of states (two) and a 9-cell neighborhood proven to support universal computation (with nite initial con guration) involves the \game of life" rule [2]. One-dimensional CA have also been shown to support universal computation [28, 27]. A recent review of theoretical results is provided in [6]. Codd proved that there does not exist a computation-universal 2-state, 5-neighbor cellular automaton with nite initial con guration. His proof concerns the original uniform model [5]. We present a universal non-uniform system embedded in such space. Section 2 presents the details of our construction, consisting of ten di erent cell rules, which are reduced to six in Section 3. Both sections involve an in nite initial con guration. Section 4 describes the implementation of a universal machine using a nite initial con guration. A quasiuniform automaton is discussed in Section 5 where quasi-uniformity implies a small number of rules distributed such that most of the grid contains one rule except for an in nitely small region which contains the others. Our nal implementationconsists of just two rules which is minimal.A discussion of our results follows in Section 6 linking them with Arti cial Life issues.
2 A universal 2-state, 5-neighbor non-uniform cellular automaton In order to prove that a two-dimensional CA is computation-universal we proceed along the lines of [2, 21, 1, 5, 31] and implement the following components4 : 1. Signals and signal pathways (wires). We must show how signals can be made to turn corners, to cross and to fan out. 2. A functionally complete set of logical gates. A set of operations is said to be functionally complete (or universal) if and only if every switching function can be expressed entirely by means of CA class numbers are those introduced by [32]. The neighborhood consists of the cell itself together with its four immediate nondiagonal neighbors. This is also the neighborhood used throughout this paper. 3 A con guration is an assignment of non-zero states to cells in the space. 4 Another approach is one in which a row of cells simulates the squares on a Turing machine tape while at the same time simulating the head of the Turing machine. This has been applied to one-dimensional CA [28, 27, 1]. 1
2
2
operations from this set [11]. We shall use the NAND function (gate) for this purpose (this gate comprises a functionally complete set and is used extensively in VLSI since transistor switches are inherently inverters [19]). 3. A clock that generates a stream of pulses at regular intervals. 4. Memory. In the following sections we describe the implementations of the above components (note: the terms `gate' and `cell' are used interchangeably). Name
Rule
Right propagation cell
x
Left propagation cell
Up propagation cell
Down propagation cell
NAND Cell
Symbol
7!
x
!
x
7!
x
x
Exclusive Or (XOR) cell (type b)
7!
x
#
Exclusive Or (XOR) cell (type d)
y
7!
x y
y
Each rule speci es the new state of the central cell. `' represents the set of states: f0; 1g. x,y 2 f0; 1g. `' is the XOR function (modulo-2 addition), `j' is the NAND function.
Table 1. Cell types (rules).
3
7!
x y
y
7!
x y
x
7!
x y
7!
x y
7!
x
x
x
No Change (NC) cell
y
x
Exclusive Or (XOR) cell (type c)
j
Symbol
"
y
x
x
x
Exclusive Or (XOR) cell (type a)
7!
x
Rule
Name
2.1
Signals and wires
A wire in our system consists of a path of connected propagation cells each containing one of the propagation rules. A signal consists of a state, or stream of states being propagated along the wire. There are four propagation cell types (i.e. four di erent rules), one for each direction: right, left, up, down (Table 1). Figure 1a shows a wire constructed from propagation cells. Figures 1b - 1d demonstrate signal propagation along the wire. Note that all cells which are not part of the machine (i.e. its components) contain the NC (No Change) rule (Table 1) which simply preserves its initial state inde nitely.
" !!!!!!!!!!!!!
"
!!!!!!!!!!!
#
#
" !!!!!!!!!!!!!
(a) cellular arrangement
"
!!!!!!!!
#
#
#
1 !!!!!!!!!!!!! 0 1 #
#
#
" !!!!!!!!!!!!!
(c) propagation of signal: time = 11
(b) propagation of signal: time = 0
"
!!!!!!!!!!!
!!!!!!!!!!!
#
1 0 1
(d) propagation of signal: time = 27
1 0 1
Fig. 1. Signal propagation along a wire. A wire in our system possesses a distinct direction, a characteristic which is highly desirable as it simpli es signal propagation [5]. In most cases signals must propagate in one direction only, and should bi-directional propagation be required then two parallel wires in opposite directions may be used. We note in Figure 1 that wires support signal propagation across corners. Fan out of signals is also straightforward as evident in Figure 2. The last problem we must address concerning signals is wire crossing (there are four possible crossings since wires may run in four directions). We rst demonstrate that at least three gates 4
" !!!!!!!!!!!!!!
!!!!!!!!!!!!!
# !!!!!!!!!!!!!!
" !!!!!!!!!!!!!!
"
(a) two-way fan out
!!!!!!!!!!!!!!!!!!!!!!!!!!!
#
# !!!!!!!!!!!!!!
(b) three-way fan out
Fig. 2. Signal fan out. (cells) are required for this operation. To see this note that one gate is insucient since there are two bits of information, denoted x and y, whereas the intersection cell can only contain one bit: y
x
!
#
?
!!
#
#
Thus, either x crosses the y (vertical) line, in which case the y signal is lost, or conversely the x signal is lost. In case there are two gates then they must eventually contain x and y (otherwise information is lost). The situation is therefore as follows: y
#
x
!
x
#
y
#
!!
The x signal gets transferred, however there is no way for the y signal to get to its gate (note that the y gate must be below the x line), since this is exactly the crossing problem we are trying to solve, and we have already shown that this cannot be done using one cell (the remaining one). A similar argument holds for the reverse situation, i.e. the intersection cell contains y. We therefore conclude that at least three gates are required for the crossing operation. Towards this end we have selected the XOR cells (Table 1) since wire crossing can be implemented using a minimal number of such gates (three). An implementation of one of the four possible crossings is given in Figure 3 (the other three are derived analogously). Note that in uniform cellular automata the implementation of wires and signals is highly complicated [31, 5, 1]. The wire itself and the operations of propagation, crossing and fan out are attained using complex structures composed of several cells. The large number of possible interactions between these structures makes the design task arduous. It is important to note the direct relationship between path length and time: if two paths branch out from a common point A to points B, C respectively, and if path length AB is strictly greater 5
y
# !
x
! !
x
#
#
y
The implementation is based on the equivalences:
x (x y) y y (x y) x
The XOR gates used are type a (see Table 1).
Fig. 3. Implementation of wire crossing. than path length AC, then a signal which fans out at A will arrive at B strictly later than at C (this issue was emphasized by [5]). 2.2
Logical gates
Table 1 includes a 2-input, 1-output NAND gate (cell), which forms a functionally complete set, thereby providing us with the second component discussed above. Two neighboring cells act as inputs while the central cell acts as the gate's output. Since NAND is functionally complete other gates such as NOT, AND and OR can be constructed from it. The XOR gate is not required for completeness purposes, however we have included it since wire crossing can be implemented using a minimal number of gates (Section 2.1). Four XOR cell types (rules) are needed to implement the four possible crossings. Once all crossings are possible we only need one NAND gate since the two wire inputs can always be made to arrive at the two input cells of the gate's neighborhood5. 2.3
Clock
The third component of our system is a clock that generates a stream of pulses at regular intervals. We implement this using a wire loop, i.e. a loop of propagation cells. Figure 4 presents the implementation. Note that any desired waveform can be produced by adjusting the size and content of the loop. The implementation of the clock is made simple due to the manner in which wires are constructed, i.e. as cellular arrangements. Thus it is possible to obtain such a closed loop, which proves highly useful in our case.
3 Reducing the number of rules In Section 2 we presented the components of a universal machine employing ten di erent cell rules (Table 1). This number may be reduced to six, by using a more complex wiring scheme, involving the implementation of the propagation cells using XOR gates (implementation is not shown due to lack of space). 5
Note that both signals must arrive synchronously. This is possible since delays can always be introduced by using, e.g., loops which are feasible once crossings are implemented.
6
" !!!
"
#
"
#
# !!!!!
(a) cellular arrangement
0 1 0 1 0 1 0 1
1 0 1 0 1 0 1 0 1
0 0 0 1 0 0 0 1
1 0 0 0 1 0 0 0 1
(b) examples of pulse signals
Fig. 4. Implementation of clock.
4 Implementing a universal machine using nite con gurations The components presented in the previous sections are sucient in order to build a universal machine using an in nite initial con guration. Codd conjectured that an unbounded but boundable propagation is a necessary condition for computation universality and used this conjecture in a proof that there does not exist a 2-state, 5-neighbor universal cellular automaton with nite initial con guration [5]. Following his work universality was implemented by using more states or larger neighborhoods [1, 2, 21]. The problem with nite con gurations involving the above components is that a computation may require an arbitrary amount of space and therefore some method must exist for increasing the information storage (memory) by arbitrarily large amounts. In order to prove universality we implement Minsky's two-register universal machine, which consists of [20]: 1. A programming unit ( nite control). 2. Two potentially in nite registers. 3. The following set of instructions: { Increase the contents of a register by one. { Decrease the contents of a register by one. { Test whether the contents of a register equal zero. The nite control unit may be realized using the components described in Sections 2 and 3. The major diculty is the implementation of the in nite registers and the three operations associated with them. According to Codd's proof a single rule in 2-state, 5-neighbor cellular space is insucient since, as noted above, unbounded but boundable propagations cannot be attained [5]. While other researchers have turned to cellular spaces with more states or larger neighborhoods, our approach is based on non-uniformity. As noted above the minimal number of rules needed to implement a register is two. Indeed, we have found two such rules: one which we denote the background rule, the other being Banks' rule [1] (rules are not shown due to lack of space). The 7
implementation of a universal computer consists of a nite control unit, which occupies a nite part of the grid. All other cells contain the background rule except for two cellular columns, in nite in one direction, containing Banks' rule. These register columns originate at the upper part of the control unit and each one represents one register (Figure 5a depicts the basic machine con guration after initialization has taken place, see ahead). The cells in these columns are denoted register cells. The three register instructions are implemented as follows: at any given moment a register column consists of an in nite6 number of cells in state 1, and a nite number in state 0, occupying the bottom part of the column. The number of 0s represents the register's value. Initially, both register columns (i.e. all register cells) are transformed (from state 0) to state 1, thus setting the register's value to zero. For each column this is accomplished by setting the bottom register cell along with its left and right neighbors to 1. The two 1s on both sides act as signals which travel upward along the column, setting all its cells to 1. Figure 5a demonstrates this process at time = 4: three cells have already been transformed to 1, the fourth is currently being transformed after which the (dual) signal will continue its upward movement. The overall e ect of this process is that the value of both registers is initialized to zero. The zero test is straightforward since it involves testing only the bottom register cell: if its state is 1 the register's value is zero, otherwise it is not. Adding one to a register is achieved by setting to 1 the cell which is at distance two to the right of the bottom register cell. Figure 5b demonstrates this operation: the left grid depicts the con guration before the operation, where the register's value is 3 and the appropriate cell is set to 1 (i.e. two cells to the right of the bottom cell). The right grid depicts the e ect of the operation (i.e. the con guration after several time steps): the column's number of zeros has increased by one which means the register's value is now 4. Subtracting one from a register is done by setting to 1 both neighboring cells of the bottom register cell (Figure 5c demonstrates this operation). Thus, the registers along with their instructions have been implemented. The initial con guration of the machine is nite, since all but a nite number of cells are set to zero. The total number of rules needed to implement a universal computer equals the number of rules necessary for implementation of the nite control unit plus the two additional memory rules (background and Banks). Thus we need a total of 12 rules using our implementation of Section 2 and 8 rules using the implementation of Section 3.
5 A quasi-uniform cellular space As noted in Section 3 by increasing the complexity of our machine a reduced set of rules (six) may be used to construct the nite control. We can go one step further and construct the nite control with only one rule, e.g. the Banks rule [1] (note that this increases the complexity of the basic operations). As noted in Section 4 the complicating issue is not the nite control but rather the in nite memory which cannot be implemented in uniform 2-state, 5-neighbor cellular space. Our conclusion from the previous paragraph is that a universal computer can be implemented using only two rules: background and Banks. Thus we have implemented universal computation using the minimal number of rules. The two rules necessary to implement universal computation are distributed unevenly. Most of the grid contains the background rule, except for an in nitely small region which contains the other one. By this we mean that although there is an in nite number of both rules in the grid, each (in nite) row contains an in nite number of background rules with only a nite number of the other. In fact, except for a nite region of the grid each row contains only two Banks rules and an in nite number of background rules. Hence we say that our cellular space is quasi-uniform. 6
More precisely the number of cells in state 1 tends to in nity as time progresses, see ahead.
8
b b b b b b
b b b b
b b b b b b
b b b b
b b b b b b
b b b b b b
b r b b r b 1 r 1 b 1 b b 1 b b 1 b
b b b b b b
b b b b b b
b r b b r b 1 r 1 b 1 b b 1 b b 1 b
b b b b b b
b b b b b b
nite control
b b b b b b
b b b b
b b b b b b
b b b b
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b
(a) initial con guration: both registers set to zero.
b b b b b b
b b b b b b
b b b b b b
1 b b b 1 b b b 1 b b b
b b b b b b
b b b b b b
b b b b b
1 b b b 1 b b b 1 b b b
r b b b r b b b r b 1 b
b b b b b b
b b b b b b
b b b b b b
b b b b b b
b b b b b b
b b b b b b
1 b b b 1 b b b
r r r r
b b b b
b b b b
b b b b
b b b b r b r b
b b b b b b
b b b b b b
(b) adding one to a register.
r b b b r b b b 1 r 1 b b
1 1 1 1
(c) subtracting one from a register
b denotes a cell in state 0 containing the background rule, r denotes a cell in state 0 containing Banks' rule.
In gures (b) and (c) the left grid shows the con guration before the operation, the right grid shows the con guration upon its completion (after several time steps). The bottom line represents the bottom register cell and its neighbors.
Fig. 5. Register operation.
6 Discussion We presented a quasi-uniform 2-state, 5-neighbor cellular automaton capable of universal computation. Quasi-uniformity implies a small number of rules distributed such that most of the grid contains one rule except for an in nitely small region which contains the others. Three implementations were described in which the number of rules used is: 12, 8 and nally 2. The following paragraphs provide a discussion of a speculative nature linking our result with that 9
of Langton [12] (see also [14]). He addressed the following question: under what conditions can we expect a dynamics of information to emerge spontaneously and come to dominate the behavior of a physical system? This was studied in the context of CA where the question becomes: under what conditions can we expect a complex dynamics of information to emerge spontaneously and come to dominate the behavior of a CA? [12]. Langton showed that the rule space of CA consists of two primary regimes of rules, periodic and chaotic, separated by a transition regime. His main conclusion was that information processing can emerge spontaneously and come to dominate the dynamics of a physical system in the vicinity of a critical phase transition. According to Codd's proof a uniform (single rule) 2-state, 5-neighbor cellular space is insucient for universal computation since unbounded but boundable propagations cannot be attained [5]. Either every con guration yields an unboundable propagation or every con guration yields a bounded propagation. In the context of Langton's work bounded propagations correspond to xed point rules (class I) and unboundable propagations correspond either to periodic rules (class II) or chaotic ones (class III). Complex behavior (class IV) cannot be attained. By using a quasi-uniform, two rule, cellular space we have been able to achieve unbounded but boundable propagations, thus attaining class IV behavior. Langton suggested that the information dynamics which gave rise to life came into existence when global or local conditions brought some medium through a critical phase transition. Imagine an information-based world consisting of a uniform cellular automaton, which is not within the class IV region. If we wanted to attain class IV behavior the entire space would have to \jump", i.e. go through a phase transition, to a class IV rule (assuming this is at all possible, e.g. in our case of a 2-state, 5-neighbor space it is not). However, as a conclusion of the work presented above we o er an alternative: a small perturbation may be enough to cause some in nitely small part of the world to change. This would be sucient to induce a (possible) transition from class II or class III behavior to class IV behavior. Furthermore, such a change could be e ected upon a very simple world (in our case 2-state, 5-neighbor). As noted by [3] \frozen accidents" play an important role in the evolutionary process. These accidents are mainly caused by external conditions, i.e. external relative to a given system's laws of functioning. A (highly) tentative comparison may be drawn to the famous experiment performed by [17] (see also [18]) in which methane, ammonia, water and hydrogen, representing a possible atmosphere of the primitive Earth, were subjected to an electric spark for a week. After this period simple amino acids were found in the system. The analogy to our CA world is as follows: we start with a simple uniform world, consisting of a single rule, which does not support complex (class IV) behavior7 . At some point, a \spark" causes a perturbation in which a small number of cells change their rule. This is all that is needed. Our central conclusion is that such an in nitely small change in our world can suce to generate a phase transition such that class IV behavior becomes possible. Note that this can happen independently in other regions of the world as well. While the above discussion has been of a speculative nature we may also draw some practical conclusions from our work. As noted in Section 1 the main diculty with the CA approach seems to lie with the extreme low-level representation of the interactions. Essentially, we construct world models at the level of physics. By slightly changing the rules of the game (no pun intended) we can increase the \capacity" for AL modeling while preserving the main features of CA, namely massive parallelism, locality of cellular interactions and simplicity of cells. Thus, we obtain a simple, general model (Section 1) which allows us to evolve complex behavior with the ability to explore, in-depth, 7
either due to inability of the cellular space to support such behavior at all or due to the rule being in the non class IV regions of rule space.
10
the inner workings of the evolutionary process. We have already begun venturing along this path [24, 25]. In these works a CA derived model is explored in which the basic units are slightly more complex than those of CA. Furthermore, evolution proceeds not only in state space as in CA but also in rule space, i.e. rules evolve over time. We have observed several phenomena of interest involving emergence, evolution and multi-cellularity. Another example is that of embryonics, standing for embryological electronics [15, 16, 7]. This is a CA based approach in which three principles of natural organization are employed: multi-cellular organization, cellular di erentiation and cellular division. Their intent is to create an architecture which is complex enough for (quasi) universal computation yet simple enough for physical implementation. The approach represents another attempt at confronting the aforementioned problem of CA, namely the low level of operation. It is hoped that the development of such AL models will serve the two-fold goal of: (1) increasing our understanding of biology and (2) enhancing our understanding of arti cial models, thereby providing us with the ability to improve their performance. AL research opens new doors providing us with novel opportunities to explore issues such as adaptation, evolution and emergence which are central both in natural environments as well as man-made ones.
Acknowledgment I am grateful to Daniel Mange for his suggestions which helped improve this paper.
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